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arXiv:2609.36431

Transverse momentum as the counter-diabatic generator in bent waveguide couplers

Yu-kai Lee, Shuo-Yen Tseng

quant-phphysics.opticsSubmitted Sep 29, 2026• Announced Sep 30, 2026• Revised Oct 16, 2026 (v2)
v1 · Sep 29, 2026v2 · Oct 16, 2026
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Abstract

Bending the axis of a mode-evolution coupler suppresses the nonadiabatic coupling between its supermodes, and bent couplers of this kind were recently shown to realize the counter-diabatic (CD) protocol. Here we identify the operator responsible. Rigid lateral displacement of a waveguide structure is generated by the transverse momentum $\hat p_x$, whose diagonal matrix elements vanish in a real supermode basis and whose off-diagonal element is purely imaginary; in a two-mode system such an operator is proportional to $\sy$. The CD term of a two-waveguide coupler is itself proportional to $\sy$, while the detuning and the coupling, the two parameters set by the waveguide widths and spacing, lie in the $σ_z$--$σ_x$ plane and cannot produce it. The transverse momentum is therefore the CD generator of the bent coupler, by necessity rather than by design. Equating the term it supplies to the nonadiabatic coupling gives the axis slope in closed form as a ratio of two matrix elements of the unperturbed supermodes at a single cross section, which a commutator identity recasts in coupled-mode variables as $\dot{\xo}=\dotθ/(γ\dbeta^{2})$. The expression agrees with the numerical offset search to $0.3\%$ and requires no search. Beam propagation simulations confirm the design and compare it, for the first time, with the CD protocol realized by unitary transformation in a straight coupler built from the same reference structure. The two devices reach $0.994$ and $1.000$ supermode fidelity where the untreated coupler reaches $0.886$, and agree in bandwidth and fabrication tolerance, while their internal trajectories differ by exactly the frame transformation that relates them.

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Bending the axis of a mode-evolution coupler suppresses the nonadiabatic coupling between its supermodes, and bent couplers of this kind were recently shown to realize the counter-diabatic (CD) protocol. Here we identify the operator responsible. Rigid lateral displacement of a waveguide structure is generated by the transverse momentum p^x\hat p_xp^​x​, whose diagonal matrix elements vanish in a real supermode basis and whose off-diagonal element is purely imaginary; in a two-mode system such an operator is proportional to \sy\sy\sy. The CD term of a two-waveguide coupler is itself proportional to \sy\sy\sy, while the detuning and the coupling, the two parameters set by the waveguide widths and spacing, lie in the σzσ_zσz​--σxσ_xσx​ plane and cannot produce it. The transverse momentum is therefore the CD generator of the bent coupler, by necessity rather than by design. Equating the term it supplies to the nonadiabatic coupling gives the axis slope in closed form as a ratio of two matrix elements of the unperturbed supermodes at a single cross section, which a commutator identity recasts in coupled-mode variables as \xo˙=θ˙/(γ\dbeta2)\dot{\xo}=\dotθ/(γ\dbeta^{2})\xo˙​=θ˙/(γ\dbeta2). The expression agrees with the numerical offset search to 0.3%0.3\%0.3% and requires no search. Beam propagation simulations confirm the design and compare it, for the first time, with the CD protocol realized by unitary transformation in a straight coupler built from the same reference structure. The two devices reach 0.9940.9940.994 and 1.0001.0001.000 supermode fidelity where the untreated coupler reaches 0.8860.8860.886, and agree in bandwidth and fabrication tolerance, while their internal trajectories differ by exactly the frame transformation that relates them.
We investigate the quantum mechanics of a single particle constrained to move along an arbitrary smooth reference curve by a confinement that is allowed to vary along the waveguide. The Schr\"odinger equation is evaluated in the adapted coordinate frame and a transverse-mode decomposition is performed, taking into account both curvature and torsion effects and the possibility of a cross-section potential that changes along the curve in an arbitrary way. We discuss the adiabatic structure of the problem, and examine nonadiabatic couplings that arise due to the curved geometry, the varying transverse profile, and their interplay. The exact multimode matrix Hamiltonian is taken as the natural starting point for few-mode approximations. Such approximate equations are provided, and it is worked out how these recover known results for twisting waveguides and can be applied to other types of waveguide designs. The quantum waveguide Hamiltonian is recast into a form that clearly illustrates how it generalizes the Born-Oppenheimer Hamiltonian encountered in molecular physics. In analogy to the latter, we explore the local gauge structure inherent to the quantum waveguide problem and suggest the usefulness of diabatic states, giving an explicit construction of the adiabatic-to-diabatic basis transformation.
Summary form only given. The manipulation of the phase of classical and quantum light is a major key to quantum photonics [1]. Integrated optics has several advantages over bulk optical systems, most notably in robustness, handling and size. Writing three-dimensional waveguides circuits into transparent materials using an ultrafast laser is thereby a new emerging technology [2; 3] with possible applications in quantum communication [4], and quantum computing [5]. In our work, we experimentally demonstrate the high-precision manipulation of the optical phase of classical and quantum light, using a generalized directional coupler.The behavior of light in directional coupler consisting of two waveguides can be described by the Hamiltonian H=T+V1+V2, where T is the kinetic energy and Vi is the potential energy of waveguide i = 1 and i = 2. The wavefunction of the system is constructed as a superposition of the single waveguide modes Ψ=a|1+b|2, and the Schrödinger equation of the coupler reads as HΨ=ksmΨ, where ksm denotes the propagation constant of the supermode. The propagation constants of the supermodes can be calculated straightforwardly as ksm1,2=k0±κ(1∓ σ)/(1-σ2) , where k0 is the propagation constant of the individual waveguides, κ=2|V2|1=1|V1|2 represents the coupling constant, and σ=1|=2|1is the direct overlap of the individual waveguide modes. For σ → 0 the well known distribution of ksm1,2 is preserved. As ksm = (ksm1+ksm2)/2>k0, in a generalized directional coupler a phase delay Δφ is introduced that readswith lC = π/κ as the coupling length. The phase delay depends only on σ, which itself is a function of the spacing between the waveguides. As the spacing can be controlled in a very accurate manner, this approach allows the highly precise manipulation of the optical phase along a directional coupler. In order to measure this effect, we set up a tuned Mach-Zehnder (MZ) interferometer with identical lengths of the interferometer arms. A third waveguide was applied to one of the arms, forming an additional directional coupler in which the light completely couples forth and back with no intensity losses in the relevant interferometer arm (Fig. 1(a)). However, as the change in the propagation constant in this arm results in a phase delay, at the output of the MZ the ratio of intensities (for classical light) and the count rates (for single photons) changes. To obtain an all-reflecting coupler (ARC) for different coupling constants, the respective length of a coupler has to be adjusted and the associated phase delays are shown in Fig. 1(b). Due to decreasing σ for larger distances between the waveguides, the accumulated phase difference shrinks, even if the length of the coupler increases. Normalizing the phase difference by length allows the direct calculation of the change of the propagation constant caused by the ARC (Fig. 1(c)).In conclusion, we demonstrated a highly precise method for the manipulation of the optical phase in integrated waveguide structures by employing generalized directional coupling. The propagation constants of the supermodes in the structures are controlled by the spacing between the waveguides in the ARC. Our approach holds for classical as well as quantum light, and has the potential to find its use in a multitude of integrated-optical devices.
Third-order nonlinear processes require phase matching between the interacting fields to achieve high efficiencies. Typically in guided-wave χ(3)χ^{(3)}χ(3) platforms this is achieved by engineering the dispersion of the modes through the transverse profile of the device. However, this limits the flexibility of the phase matching that can be achieved. Instead, we analyze four-wave mixing in a pair of asymmetric waveguides and show that phasematching may be achieved in any χ(3)χ^{(3)}χ(3) waveguide by coupling of a nondegenerate pump from an adjacent waveguide. We demonstrate the additional flexibility that this approach yields in the case of photon-pair generation by spontaneous FWM, where the supermode dispersion may be modified to produce pure heralded single photons -- a critical capability required for example by silicon platforms for chip-scale quantum photonics.
The recent advent of integrated waveguide systems with reconfigurable propagation constants and coupling coefficients has opened the door to using waveguide detuning as a resource for readily tailoring the quantum properties of light states. Here we theoretically demonstrate that waveguide mode detuning can be used for molding the nonclassical properties of two interacting quantum optical fields in integrated waveguide couplers. In particular, we explore the states that are generated by conditional measurements when one of the input ports of the waveguide coupler is excited by coherent states, squeezed vacuum states, and thermal states, while the other port is excited by a single-photon Fock state. We explore the detuning range required to attain nonclassical states. Our findings could pave the way for a robust integrated-optics protocol, providing enhanced control and engineering capabilities over multiphoton quantum states.
We propose an adiabatic method for transfer of light in a three-waveguide directional coupler in a single mode. In our scheme the propagation coefficients of the two outer waveguides are identical, but may vary simultaneously along the propagation direction. In contrast to earlier work [E. Paspalakis, Opt. Commun. 258, 30 (2006)], where the coupling coefficients are designed to act in the counterintuitive order for an input in one of the outer waveguides, here we may have initially any light input in each of the waveguides and any order of coupling coefficients. Consequently, all three adiabatic eigenstates of the system are used in the dynamics. These solutions allow us to design some interesting waveguide devices.